Compute the eigenvalues of a general matrix.
a : (M, M) array_like
w : (M,) ndarray
This is a simple interface to the LAPACK routines dgeev and zgeev that sets those routines’ flags to return only the eigenvalues of general real and complex arrays, respectively.
Illustration, using the fact that the eigenvalues of a diagonal matrix are its diagonal elements, that multiplying a matrix on the left by an orthogonal matrix, Q, and on the right by Q.T (the transpose of Q), preserves the eigenvalues of the “middle” matrix. In other words, if Q is orthogonal, then Q * A * Q.T has the same eigenvalues as A:
>>> from numpy import linalg as LA >>> x = np.random.random() >>> Q = np.array([[np.cos(x), -np.sin(x)], [np.sin(x), np.cos(x)]]) >>> LA.norm(Q[0, :]), LA.norm(Q[1, :]), np.dot(Q[0, :],Q[1, :]) (1.0, 1.0, 0.0)
Now multiply a diagonal matrix by Q on one side and by Q.T on the other:
>>> D = np.diag((-1,1)) >>> LA.eigvals(D) array([-1., 1.]) >>> A = np.dot(Q, D) >>> A = np.dot(A, Q.T) >>> LA.eigvals(A) array([ 1., -1.])